PROJECT SUMMARY/ABSTRACT The Weber-Fechner law is perhaps the oldest quantitative relationship in psychology. Detailed neurophysiology of the visual system demonstrates that neural representations of extrafoveal retinal position obey Weber-Fechner scaling such that the width of the receptive ?eld centered at x goes up like x, creating a neural basis for the behavioral Weber-Fechner law. Behavioral work and cognitive modeling suggests that neural representations of other variables such as time and space in the hippocampus should also obey Weber-Fechner scaling, suggesting that Weber-Fechner scaling is a fundamental principle of brain function. Weber-Fechner scaled representations of time and space constitute a signi?cant computational challenge. A recent proposal for constructing a scale- invariant representations of time and one-dimensional space utilizes the Laplace transform of the past as an intermediate representation. This framework has been shown to be roughly consistent with neurophysiological recordings from hippocampal time cells and place cells. Cognitive models built from this representation can account for a broad range of behavioral ?ndings from a variety of different types of memory tasks. Access to the Laplace transform enables rapid and ?exible computation; preliminary theoretical work suggests a mapping hypothesis between translation of a function de?ned over a Weber-Fechner scaled domain and hippocampal theta oscillations. The proposed research builds towards a general theoretical framework for neural representations of time, space and number. One subproject works to test quantitative predictions of the existing mathematical framework by developing new data analysis tools. These tools will be evaluated on existing ensemble recordings from ro- dent and primate hippocampus and medial prefrontal cortex made available through cooperative agreement with leading neurophysiologists. A second subproject works to establish the potential neurophysiological basis of the equations by building detailed circuit models to implement the equations. These circuit models will draw on known properties from both in vitro and in vivo neurophysiology. A third subproject extends the mathematical basis of the framework from 1-dimesional representations of time, space, and number to more general N -dimensional problems. This approach will derive translation operators and control circuits for translation along more general paths using an approach inspired by differential geometry. Taken together, the three subprojects could result in a substantial step towards a general quantitative neural theory for many aspects of memory and ?exible neural computation.